Is there a way to move, scale and duplicate hidden objects with the gizmo?
Summary
You can constrain the location, scale and rotation of the to-be-hidden object to another object, which you then manipulate to indirectly influence the invisible object.

Preface
Some terminology I'm using for clarification:
'Target' = The object that Boolean-intersects the mesh, you want to hide this but manipulate it as well;
'Modifier' = The object that you need to manipulate to indirectly manipulate the Target object.

Constraining the Target to the Modifier object
Create a duplicate of the Target object (linked(Alt-D) if you also want to edit the Target's mesh, but a link is not required).
This duplicate will be the object you manipulate, the Modifier object.   

Note that you can also just use an Empty or generally any object; I like to duplicate the object to better see what is going on.

Create three constraints on the Target object; Copy Location, Copy scale and Copy Rotation.
For Copy Location you will want to enable the Offset option; this will allow you to move the Target object away from the Modifier object instead of them occupying the exact same point in space.

Copy Scale and Copy Rotation should be set up correctly as soon as you add the constraint. You may need to adjust the 'Invert' options on the Copy Rotation constraint to get the Target to link in the correct orientation.

The target for all constraints is the Modifier object.

If you select both the Modifier and Target object and duplicate these(Shift-D), the Target copy will already have it's constraints set up to target the duplicated Modifier. 



Images for reference
Image 1: Here you can see the setup of objects and the constraints. The Target is already hidden and is selected in the Outliner. Note the position of the hole in the Cube.


Image 2: Now if you move/rotate/scale the Modifier object, the Target will move/rotate/scale accordingly.